McKay quiver of a group in nice characteristic

Spectrum of a McKay graph

Let $𝑊$ be a $𝕂 [𝐺]$ -module affording character $𝜂$ and $Γ 𝑊 (𝐺)$ be its McKay quiver with adjacency matrix $𝐴 𝑊 (𝐺) = (𝑎 𝑖 𝑗)$ . Then the eigenvectors of $𝐴 𝑊 (𝐺)$ are the columns of the character table for $𝐺$ , and the eigenvalues are the corresponding values of $𝜂$ . #m/thm/rep2

Proof

Let $𝜂$ be the character afforded by $𝑊$ . For each $𝑖 \in ℕ 𝑟$ , we have

𝑟 \sum 𝑗 = 1 𝑎 𝑖 𝑗 𝜒 𝑗 = 𝜂 𝜒 𝑖

and thus for any $𝑔 \in 𝐺$ ,

𝐴 𝑊 (𝐺) ⎡ ⎢ ⎢ ⎣ 𝜒 1 (𝑔) ⋮ 𝜒 𝑟 (𝑔) ⎤ ⎥ ⎥ ⎦ = 𝜂 (𝑔) ⎡ ⎢ ⎢ ⎣ 𝜒 1 (𝑔) ⋮ 𝜒 𝑟 (𝑔) ⎤ ⎥ ⎥ ⎦

which gives $𝑟$ linearly independent eigenvectors and is thus complete.

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